Towards an unified theory for testing statistical hypothesis: Multinormal mean with nuisance covariance matrix

TL;DR

This paper proposes a unified framework for testing hypotheses about the mean in a multinormal distribution with unknown covariance, integrating Bayesian, frequentist, and Fisherian paradigms using Fisher's fiducial inference and Wald's decision theory.

Contribution

It introduces a novel approach combining Fisher's fiducial inference and Wald's decision theory to unify different statistical testing paradigms for multinormal means with nuisance covariance.

Findings

Tests based on the union-intersection principle are effective.

The framework achieves reconciliation of multiple statistical paradigms.

The approach handles arbitrary unknown covariance matrices.

Abstract

Under a multinormal distribution with arbitrary unknown covariance matrix, the main purpose of this paper is to propose a framework to achieve the goal of reconciliation of Bayesian, frequentist and Fisherian paradigms for the problems of testing mean against restricted alternatives (closed convex cones). Combining Fisher's fiducial inference and Wald's decision theory via d-admissibility into an unified approach, the goal can then be achieved. To proceed, the tests constructed via the union-intersection principle are studied.